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Why do people saw Dilithium is a fuel? All it seems to do is catalyzes the M-AM reactions.
Canonically is there any dialogue or graphic saying Dilithium or antimatter was used?

I don't think so. The only technical bit I remember is:

LAFORGE (OC): Doctor!COCHRANE: Yeah.LAFORGE: Would you mind taking a look at this?COCHRANE: Yeah.LAFORGE: I've tried to reconstruct the intermix chamber from what I remember at school. Tell me if I got it right.COCHRANE: School? You learned about this in school?LAFORGE: Oh yeah. 'Basic Warp Design' is a required course at the Academy. The first chapter is called 'Zefram Cochrane'.COCHRANE: Well, it looks like you got it right.BARCLAY: Commander. This is what we're thinking of using to replace the damaged warp plasma conduit.
(Geordi uses his ocular implant to examine a spiral of copper tubing)LAFORGE: Yeah. Yeah, that's good, but you need to reinforce this copper tubing with a nano-polymer.

I don't think there is much else.

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LAFORGE: The hard part's going to be calibrating the thermal curve necessary to start a controlled reaction.
RIKER: Assuming you can, can you regulate the reaction?
WESLEY: There's just enough crystal to do it. We plan to channel the reaction through the chips.
LAFORGE: Are we good?
RIKER: You're better than good. Great. Brilliant. It's going to be fun. Carry on.

That seems to suggest that the function of the dilithium crystals is to somehow regulate the matter/antimatter reaction, rather than to provide the fuel for it.

Personally, I like to think that Cochrane used "simple impulse" power to obtain warp drive for the Phoenix's brief test flight. It's not like he was planning on going all the way to Alpha Centauri right then and there, IMO. I think he just needed something that could get the ship to warp one for a few minutes.

__________________"Don't sweat the small stuff--it makes you small-minded..."

LAFORGE: The hard part's going to be calibrating the thermal curve necessary to start a controlled reaction.
RIKER: Assuming you can, can you regulate the reaction?
WESLEY: There's just enough crystal to do it. We plan to channel the reaction through the chips.
LAFORGE: Are we good?
RIKER: You're better than good. Great. Brilliant. It's going to be fun. Carry on.

That seems to suggest that the function of the dilithium crystals is to somehow regulate the matter/antimatter reaction, rather than to provide the fuel for it.

I have always gotten this impression as well.

The question I have is, was there any indication that Cochrane used a matter/anti-matter reaction that first time?

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Well in Star Trek IV when they time travelled Scotty said it drained the dilithium crystals and they had to recrystalize them by using photons from a nuclear reactor. He clearly said they couldn't go to warp without them.

Well in Star Trek IV when they time travelled Scotty said it drained the dilithium crystals and they had to recrystalize them by using photons from a nuclear reactor. He clearly said they couldn't go to warp without them.

I always saw that as the trip overworked the dilithium crystals, causing them to deteriorate to the point where they were no longer usable.

__________________"Don't sweat the small stuff--it makes you small-minded..."

Well in Star Trek IV when they time travelled Scotty said it drained the dilithium crystals and they had to recrystalize them by using photons from a nuclear reactor. He clearly said they couldn't go to warp without them.

I always saw that as the trip overworked the dilithium crystals, causing them to deteriorate to the point where they were no longer usable.

My biggest issue regarding First Warp Flight isent dilithium but the manner how it was acthived. I didnt like how Cochrane was proytrayed as a drunk and i wasent sold how he had recources to plan this flight in middle of Montana in some old nucular hangout. Going warp should be really hard and dangerous and first flight should really be unmanned and not manned like Phoenix.